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Certainly! Here is a detailed step-by-step solution to your assignment problem regarding the monthly salary data:
### Given Salaries:
580, 620, 700, 830, 910, 620, 700, 830, 860, 1020
### Step 1: Calculate the Range
The range of a data set is calculated as the difference between the maximum and minimum values.
- Minimum salary: 580
- Maximum salary: 1020
[tex]\[ \text{Range} = \text{Maximum} - \text{Minimum} = 1020 - 580 = 440 \][/tex]
### Step 2: Calculate the Mean Salary
The mean salary is the average of all the salaries.
[tex]\[ \text{Mean Salary} = \frac{\sum \text{Salaries}}{\text{Number of Salaries}} \][/tex]
Summing up all the salaries:
[tex]\[ 580 + 620 + 700 + 830 + 910 + 620 + 700 + 830 + 860 + 1020 = 7670 \][/tex]
There are 10 salaries in total:
[tex]\[ \text{Mean Salary} = \frac{7670}{10} = 767.0 \][/tex]
### Step 3: Calculate the Standard Deviation of the Salaries
Standard deviation measures the dispersion or spread of salaries around the mean. It is calculated using the formula:
[tex]\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \][/tex]
Where:
- [tex]\(x_i\)[/tex] are the individual salary values.
- [tex]\(\mu\)[/tex] is the mean salary.
- [tex]\(N\)[/tex] is the total number of salaries.
First, calculate the variance:
[tex]\[ \text{Variance} = \frac{\sum (x_i - \mu)^2}{N} \][/tex]
Using the mean of 767.0, we find:
[tex]\[ \begin{aligned} (580 - 767.0)^2 = 34969.0 \\ (620 - 767.0)^2 = 21609.0 \\ (700 - 767.0)^2 = 4489.0 \\ (830 - 767.0)^2 = 3969.0 \\ (910 - 767.0)^2 = 20449.0 \\ (620 - 767.0)^2 = 21609.0 \\ (700 - 767.0)^2 = 4489.0 \\ (830 - 767.0)^2 = 3969.0 \\ (860 - 767.0)^2 = 8649.0 \\ (1020 - 767.0)^2 = 63889.0 \\ \end{aligned} \][/tex]
Summing these squared differences:
[tex]\[ 34969.0 + 21609.0 + 4489.0 + 3969.0 + 20449.0 + 21609.0 + 4489.0 + 3969.0 + 8649.0 + 63889.0 = 188090.0 \][/tex]
Now, divide by the number of salaries (10):
[tex]\[ \text{Variance} = \frac{188090.0}{10} = 18809.0 \][/tex]
Finally, the standard deviation:
[tex]\[ \sigma = \sqrt{18809.0} \approx 137.19 \][/tex]
### Step 4: Increase Each Salary by 20% and Calculate New Mean and Standard Deviation
To find the new salaries after a 20% increase:
[tex]\[ \begin{aligned} 580 \times 1.2 = 696 \\ 620 \times 1.2 = 744 \\ 700 \times 1.2 = 840 \\ 830 \times 1.2 = 996 \\ 910 \times 1.2 = 1092 \\ 620 \times 1.2 = 744 \\ 700 \times 1.2 = 840 \\ 830 \times 1.2 = 996 \\ 860 \times 1.2 = 1032 \\ 1020 \times 1.2 = 1224 \\ \end{aligned} \][/tex]
### New Mean Salary:
[tex]\[ \text{New Mean Salary} = \frac{\sum \text{Increased Salaries}}{\text{Number of Salaries}} \][/tex]
Summing the new salaries:
[tex]\[ 696 + 744 + 840 + 996 + 1092 + 744 + 840 + 996 + 1032 + 1224 = 9204 \][/tex]
Divide by the number of salaries (10):
[tex]\[ \text{New Mean Salary} = \frac{9204}{10} = 920.4 \][/tex]
### New Standard Deviation:
First, calculate the variance for the increased salaries using the new mean of 920.4:
[tex]\[ \begin{aligned} (696 - 920.4)^2 = 50193.76 \\ (744 - 920.4)^2 = 31032.96 \\ (840 - 920.4)^2 = 6465.16 \\ (996 - 920.4)^2 = 5718.76 \\ (1092 - 920.4)^2 = 29447.36 \\ (744 - 920.4)^2 = 31032.96 \\ (840 - 920.4)^2 = 6465.16 \\ (996 - 920.4)^2 = 5718.76 \\ (1032 - 920.4)^2 = 12482.56 \\ (1224 - 920.4)^2 = 91871.36 \\ \end{aligned} \][/tex]
Summing these squared differences:
[tex]\[ 50193.76 + 31032.96 + 6465.16 + 5718.76 + 29447.36 + 31032.96 + 6465.16 + 5718.76 + 12482.56 + 91871.36 = 271429.04 \][/tex]
Divide by the number of salaries (10):
[tex]\[ \text{Variance} = \frac{271429.04}{10} = 27142.904 \][/tex]
Finally, the new standard deviation:
[tex]\[ \sigma = \sqrt{27142.904} \approx 164.63 \][/tex]
### Summary:
- Range: 440
- Mean Salary: 767.0
- Standard Deviation: 137.19
- New Mean Salary (after 20% increase): 920.4
- New Standard Deviation (after 20% increase): 164.63
### Given Salaries:
580, 620, 700, 830, 910, 620, 700, 830, 860, 1020
### Step 1: Calculate the Range
The range of a data set is calculated as the difference between the maximum and minimum values.
- Minimum salary: 580
- Maximum salary: 1020
[tex]\[ \text{Range} = \text{Maximum} - \text{Minimum} = 1020 - 580 = 440 \][/tex]
### Step 2: Calculate the Mean Salary
The mean salary is the average of all the salaries.
[tex]\[ \text{Mean Salary} = \frac{\sum \text{Salaries}}{\text{Number of Salaries}} \][/tex]
Summing up all the salaries:
[tex]\[ 580 + 620 + 700 + 830 + 910 + 620 + 700 + 830 + 860 + 1020 = 7670 \][/tex]
There are 10 salaries in total:
[tex]\[ \text{Mean Salary} = \frac{7670}{10} = 767.0 \][/tex]
### Step 3: Calculate the Standard Deviation of the Salaries
Standard deviation measures the dispersion or spread of salaries around the mean. It is calculated using the formula:
[tex]\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \][/tex]
Where:
- [tex]\(x_i\)[/tex] are the individual salary values.
- [tex]\(\mu\)[/tex] is the mean salary.
- [tex]\(N\)[/tex] is the total number of salaries.
First, calculate the variance:
[tex]\[ \text{Variance} = \frac{\sum (x_i - \mu)^2}{N} \][/tex]
Using the mean of 767.0, we find:
[tex]\[ \begin{aligned} (580 - 767.0)^2 = 34969.0 \\ (620 - 767.0)^2 = 21609.0 \\ (700 - 767.0)^2 = 4489.0 \\ (830 - 767.0)^2 = 3969.0 \\ (910 - 767.0)^2 = 20449.0 \\ (620 - 767.0)^2 = 21609.0 \\ (700 - 767.0)^2 = 4489.0 \\ (830 - 767.0)^2 = 3969.0 \\ (860 - 767.0)^2 = 8649.0 \\ (1020 - 767.0)^2 = 63889.0 \\ \end{aligned} \][/tex]
Summing these squared differences:
[tex]\[ 34969.0 + 21609.0 + 4489.0 + 3969.0 + 20449.0 + 21609.0 + 4489.0 + 3969.0 + 8649.0 + 63889.0 = 188090.0 \][/tex]
Now, divide by the number of salaries (10):
[tex]\[ \text{Variance} = \frac{188090.0}{10} = 18809.0 \][/tex]
Finally, the standard deviation:
[tex]\[ \sigma = \sqrt{18809.0} \approx 137.19 \][/tex]
### Step 4: Increase Each Salary by 20% and Calculate New Mean and Standard Deviation
To find the new salaries after a 20% increase:
[tex]\[ \begin{aligned} 580 \times 1.2 = 696 \\ 620 \times 1.2 = 744 \\ 700 \times 1.2 = 840 \\ 830 \times 1.2 = 996 \\ 910 \times 1.2 = 1092 \\ 620 \times 1.2 = 744 \\ 700 \times 1.2 = 840 \\ 830 \times 1.2 = 996 \\ 860 \times 1.2 = 1032 \\ 1020 \times 1.2 = 1224 \\ \end{aligned} \][/tex]
### New Mean Salary:
[tex]\[ \text{New Mean Salary} = \frac{\sum \text{Increased Salaries}}{\text{Number of Salaries}} \][/tex]
Summing the new salaries:
[tex]\[ 696 + 744 + 840 + 996 + 1092 + 744 + 840 + 996 + 1032 + 1224 = 9204 \][/tex]
Divide by the number of salaries (10):
[tex]\[ \text{New Mean Salary} = \frac{9204}{10} = 920.4 \][/tex]
### New Standard Deviation:
First, calculate the variance for the increased salaries using the new mean of 920.4:
[tex]\[ \begin{aligned} (696 - 920.4)^2 = 50193.76 \\ (744 - 920.4)^2 = 31032.96 \\ (840 - 920.4)^2 = 6465.16 \\ (996 - 920.4)^2 = 5718.76 \\ (1092 - 920.4)^2 = 29447.36 \\ (744 - 920.4)^2 = 31032.96 \\ (840 - 920.4)^2 = 6465.16 \\ (996 - 920.4)^2 = 5718.76 \\ (1032 - 920.4)^2 = 12482.56 \\ (1224 - 920.4)^2 = 91871.36 \\ \end{aligned} \][/tex]
Summing these squared differences:
[tex]\[ 50193.76 + 31032.96 + 6465.16 + 5718.76 + 29447.36 + 31032.96 + 6465.16 + 5718.76 + 12482.56 + 91871.36 = 271429.04 \][/tex]
Divide by the number of salaries (10):
[tex]\[ \text{Variance} = \frac{271429.04}{10} = 27142.904 \][/tex]
Finally, the new standard deviation:
[tex]\[ \sigma = \sqrt{27142.904} \approx 164.63 \][/tex]
### Summary:
- Range: 440
- Mean Salary: 767.0
- Standard Deviation: 137.19
- New Mean Salary (after 20% increase): 920.4
- New Standard Deviation (after 20% increase): 164.63
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